Mathematics 309

Linear approximations

In 1D

If f(x) is a reasonable function of one variable,
then its graph looks linear when you look only
at small parts of it.

Of course, the line it looks like is the tangent line.

In other words, for small values of
and an arbitrary value of x0 we have an approximate equality

f(x0 + ) ~ f(x0) + f'(x0)

We can understand this a bit better if we look at an example, say f(x) = x2. Then
we have exactly

f(x0 + ) = f(x0) + 2x0
+ 2

and since f'(x) = x2 x, this means that the approximation by
the tangent line
is one which ignores the term 2.
In general the linear approximation means ignoring powers
k where k
is 2 or more. Of course if is small,
then these will be an order of magnitude smaller.